A particle P is moving along a straight line through the fixed point O.The displacement, s metres, of P from O at time t seconds is given bys = t 3 – 27t + 55 t . 0(a) Write down the distance, in metres, of P from O when t = 0(1)(b) Find an expression, in terms of t, for the velocity, v m/s, of P at time t seconds.(2)(c) Find the value of t when P is closest to O.(2)(d) Find the distance, in metres, of P from O when P is closest to O.(1)(e) Find the distance, in metres, travelled by P in the interval 0 - t - 5(3)..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
Question
A particle P is moving along a straight line through the fixed point O.The displacement, s metres, of P from O at time t seconds is given bys = t 3 – 27t + 55 t . 0(a) Write down the distance, in metres, of P from O when t = 0(1)(b) Find an expression, in terms of t, for the velocity, v m/s, of P at time t seconds.(2)(c) Find the value of t when P is closest to O.(2)(d) Find the distance, in metres, of P from O when P is closest to O.(1)(e) Find the distance, in metres, travelled by P in the interval 0 - t - 5(3)..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
Solution
(a) When t = 0, the displacement s = (0)^3 - 27(0) + 55(0) = 0 metres. So, the distance of P from O is 0 metres.
(b) The velocity v of P at time t seconds is the derivative of the displacement function with respect to time. So, v = ds/dt = 3t^2 - 27.
(c) The particle P is closest to O when its velocity is 0. So, set v = 0 and solve for t: 0 = 3t^2 - 27 => t^2 = 27/3 => t = sqrt(9) => t = 3 seconds.
(d) The distance of P from O when P is closest to O is the displacement at t = 3 seconds. So, s = (3)^3 - 27(3) + 55(3) = 27 - 81 + 165 = 111 metres.
(e) The distance travelled by P in the interval 0 ≤ t ≤ 5 is the integral of the absolute value of the velocity function from 0 to 5. So, ∫ |3t^2 - 27| dt from 0 to 5. This integral needs to be calculated numerically.
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